One disadvantage to solving systems using substitution is that isolating a variable often involves dealing with messy fractions. There is another method for solving systems of equations: the addition/subtraction method.
In the addition/subtraction method, the two equations in the system are added or subtracted to create a new equation with only one variable. In order for the new equation to have only one variable, the other variable must cancel out. In other words, we must first perform operations on each equation until one term has an equal and opposite coefficient as the corresponding term in the other equation.
We can produce equal and opposite coefficients simply by multiplying each equation by an integer. Observe:
Example 1: Add and subtract to create a new equation with only one variable:
2x + 4y | = | 3 | |
x + 3y | = | 13 |
2x + 4y | = | 4 | |
-2x - 6y | = | -26 |
4x - 2y | = | 16 | |
7x + 3y | = | 15 |
12x - 6y | = | 48 | |
14x + 6y | = | 30 |
We can add and subtract equations by the addition property of equality--since the two sides of one equation are equivalent, we can add one to one side of the second equation and the other to the other side.
Here are the steps to solving systems of equations using the addition/subtraction method:
Example 1: Solve the following system of equations:
2y - 3x | = | 7 | |
5x | = | 4y - 12 |
Example 2: Solve the following system of equations:
4y - 5 | = | 20 - 3x | |
4x - 7y + 16 | = | 0 |
Example 3: Solve the following system of equations:
2x - 5y | = | 15 | |
10y | = | 20 + 4x |
Example 4: Solve the following system of equations:
6x + 14y | = | 16 | |
24 - 9x | = | 21y |